<html>

<head>
<meta http-equiv="Content-Language" content="en-us">
<meta name="GENERATOR" content="Microsoft FrontPage 5.0">
<meta name="ProgId" content="FrontPage.Editor.Document">
<meta http-equiv="Content-Type" content="text/html; charset=windows-1252">
<title>GEO_ROT(ENT,EXI,A,B,ANG,BASIS=BAS)</title>
</head>

<body>

<p>We now describe the call <font color="#FF00FF"><b>
GEO_ROT(E,EP,A,AP,ALPHA,BASIS=B): <a href="programs/z_geo_rot1.f90">Example 
Program</a></b></font></p>

<p><a target="_blank" href="geo_rot.jpg">
<img border="0" src="geo_rot.jpg" width="562" height="404" align="left"></a></p>
<p>In the  figure (click on it for magnified view), the frame <b>g </b>represent the absolute frame of 
reference of space. The components of the vectors <b>g</b><sub>1</sub>,<b>g</b><sub>2</sub> 
and <b>g</b><sub>3</sub> are:</p>
<p align="center"><font size="4"><b>g</b><sub>1</sub>=(1,0,0) <b>g</b><sub>2</sub>=(0,1,0)
<b>g</b><sub>3</sub>=(0,0,1) </font></p>
<p align="left">The vector&nbsp; <b>A</b> are expressed in the basis (<b>e</b><sub>1</sub>,<b>e</b><sub>2</sub>,<b>e</b><sub>3</sub>). 
These vectors can be expressed in the basis (<b>g</b><sub>1</sub>,<b>g</b><sub>2</sub>,<b>g</b><sub>3</sub>) 
using a matrix E:</p>
<p align="center"><font color="#FF0000"><b><font size="5">e</font></b><font size="5"><sub>i</sub>=E<sub>ij</sub><b>g</b></font></font><sub><font color="#FF0000" size="5">j</font><font size="5">
</font></sub><font size="5">(summed over j )&nbsp;  </font></p>
<p align="center"><font size="5">or we can also write
<font color="#FF0000"><b>e</b>=E<b>g </b></font></font></p>
<p align="left"><font color="#FF0000"><b><font size="5">g</font> </b></font>and<b><font color="#FF0000">
<font size="5">e</font></font></b> are a column of 3 vectors. </p>
<p align="left">&nbsp;</p>
<p align="left">The component of an arbitrary vector <b><font color="#006600">A</font></b> 
are V<sub>i </sub>are expressed in the basis <font color="#FF0000"><b>e</b></font>. 
Thus we have:</p>
<p align="center"><font color="#006600"><b><font size="6">A </font></b></font>
<font size="6">=&nbsp; V<sub>i </sub><font color="#FF0000"><b>e<sub>i </sub></b>
</font>= V<sub>i </sub></font><b><font size="6"><font color="#FF0000">E<sub>ij</sub></font>g<sub>j</sub> 
or </font></b><font color="#006600"><b><font size="6">A </font></b></font>
<font size="6">= V<sup>t</sup></font><b><font size="6" color="#FF0000">E</font><font size="6">g</font></b></p>
<p align="left">The rotation <b>R(<font face="Symbol">a</font>)=R<sub>z</sub>(</b><font face="Symbol">a</font><sub>3</sub><b>)R<sub>y</sub>(</b><font face="Symbol">a</font><sub>2</sub><b>)R<sub>x</sub>(</b><font face="Symbol">a</font><sub>1</sub><b>)</b> 
is defined in the <b>b</b> frame: <b><font color="#0033CC" size="5">b=B</font><font size="5">g</font></b></p>
<p align="left"><font size="5"><b>Goal: Rotate the frame <font color="#FF0000">E</font> 
and the vector A attached to it with a rotation </b></font><b><font size="5">R(<font face="Symbol">a</font>) 
expressed in a factored form in the standard PTC order and defined in the frame
<font color="#0033CC">B</font>. Typically the frames <font color="#FF0000">E</font> 
and <font color="#0033CC">B</font> are attached to the body of a magnet or a 
fibre. Find the new component of the basis <font color="#FF0000">e</font>,
<font color="#FF0000">E'</font> , and the component </b><font color="#006600">A'</font><b>of 
the vector <font color="#006600">A'</font> in the global basis g.</b></font></p>
<p align="left"><font size="5"><b>Solution for </b></font><b>
<font color="#FF0000" size="5">E'</font></b><font size="5"><b>: </b></font></p>
<p align="left"><b><font color="#006600" size="6">A' </font></b><font size="6">=</font><font size="5"><b>
</b></font><b><font size="6">R(<font color="#006600">A</font>)<font color="#006600">
</font></font></b><font size="6">=&nbsp; </font><b><font size="6">R(</font></b><font size="6">V<sub>i
</sub><font color="#FF0000"><b>e<sub>i</sub></b></font></font><b><font size="6">)
</font></b><font size="6"><b>= </b></font><b><font size="6">R(</font></b><font size="6">V<sub>i
</sub></font><b><font size="6" color="#FF0000">E<sub>ij</sub></font><font size="6">g<sub>j</sub>) 
= </font></b><font size="6"><b>&nbsp;</b></font><b><font size="6">R(</font></b><font size="6">V<sup>t</sup><sub>
</sub></font><b><font size="6" color="#FF0000">E</font><font color="#0033CC" size="6">B<sup>t</sup>b</font><font size="6">)
</font></b></p>
<p align="left"><font size="6"><b>= </b>V<sup>t</sup><sub> </sub></font><b>
<font size="6" color="#FF0000">E</font><font color="#0033CC" size="6">B<sup>t</sup></font><font size="6">R</font><font color="#0033CC" size="6">b</font><font size="6"> 
= </font></b><font size="6">V<sup>t</sup><sub> </sub></font><b>
<font size="6" color="#FF0000">E</font><font color="#0033CC" size="6">B<sup>t</sup></font><font size="6">R</font><font color="#0033CC" size="6">B</font><font size="6">g 
=&gt; </font><font size="7" color="#FF0000">E'=E</font><font color="#0033CC" size="7">B<sup>t</sup></font><font size="7">R</font><font color="#0033CC" size="7">B</font></b></p>
<p align="left"><font size="5"><b>Solution for </b></font>
<font color="#006600" size="5">A<b>'</b></font><b><font size="5"> using </font>
</b><font size="5">A<sup>t </sup>= V<sup>t</sup><sub> </sub></font><b>
<font size="5" color="#FF0000">E </font></b><font size="5"><b>: </b></font></p>
<p align="left"><b><font color="#006600" size="6">A' </font></b><font size="6">= 
V<sup>t</sup><sub> </sub></font><b><font size="6" color="#FF0000">E</font><font color="#0033CC" size="6">B<sup>t</sup></font><font size="6">R</font><font color="#0033CC" size="6">B</font><font size="6">g 
= </font></b><font color="#006600" size="6">A<sup>t</sup></font><b><font color="#0033CC" size="6">B<sup>t</sup></font><font size="6">R</font><font color="#0033CC" size="6">B</font><font size="6">g 
=&gt;&nbsp;&nbsp; </font></b><font color="#006600" size="7">A<b>'</b></font><b><font size="7"><sup>t</sup>=
</font></b><font color="#006600" size="7">A<sup>t</sup></font><b><font color="#0033CC" size="7">B<sup>t</sup></font><font size="7">R</font><font color="#0033CC" size="7">B</font></b></p>
<p align="left">&nbsp;</p>
<p align="left">&nbsp;</p>
<p align="center">&nbsp;</p>
<p align="left"><b><font size="5">&nbsp;</font></b></p>
<p align="left">&nbsp;</p>
<p align="left">&nbsp;</p>
<p align="left">&nbsp;</p>
<p align="left">&nbsp;</p>

</body>

</html>